Diederik P. Kingma, Jimmy Lei Ba
6 min
Abstract
We introduce Adam, an algorithm for first-order gradient-based optimization of stochastic objective functions, based on adaptive estimates of lower-order moments. The method is straightforward to implement, is computationally efficient, has little memory requirements, is invariant to diagonal rescaling of the gradients, and is well suited for problems that are large in terms of data and/or parameters. The method is also appropriate for non-stationary objectives and problems with very noisy and/or sparse gradients. The hyper-parameters have intuitive interpretations and typically require little tuning. Some connections to related algorithms, on which Adam was inspired, are discussed. We also analyze the theoretical convergence properties of the algorithm and provide a regret bound on the convergence rate that is comparable to the best known results under the online convex optimization framework. Empirical results demonstrate that Adam works well in practice and compares favorably to other stochastic optimization methods. Finally, we discuss AdaMax, a variant of Adam based on the infinity norm.
Alex: That's clever. But you mentioned early stalls from zero starts—how does the fix dodge that?
Sam: Without it, high decay rates heavily discount new gradients, underestimating everything at first. The correction mathematically unbiased the estimates from step one, preventing timid crawls or explosions in sparse cases.
Alex: Does theory back convergence despite the noise?
Sam: Yes—an O(sqrt(T)) regret bound matching the best, improving to log(d) sqrt(T) for sparse features. The paper tests this on real datasets.
Alex: How does it play out, like on handwritten digits?
Sam: First, logistic regression on MNIST pixels. Adam converges as fast as top SGD versions, quicker than AdaGrad, dropping training cost steadily.
Alex: Makes sense for dense data. But sparse is Adam's strength—what's the test?
Sam: IMDB movie reviews as bag-of-words vectors—super sparse. With 50% dropout noise, Adam matches AdaGrad's fast drop, way better than plain SGD.
Alex: Dropout zeros half the features each time, making gradients noisier. Why does Adam handle that better than RMSProp?
Sam: RMSProp lacks the bias fix, so early steps explode with high decay for sparsity. Adam stays stable, converging where RMSProp fails.
Alex: Does this hold in deeper nets?
Sam: Yep—in multi-layer nets with dropout, Adam beats memory-hungry quasi-Newton methods in speed and reliability. On convolutional nets for vision, it works well despite wild layer gradients, needing few tweaks like a default learning rate of 0.001.
Alex: What about CIFAR-10 color images?
Sam: Conv net with dropout—Adam drops cost fast early, then pulls ahead of AdaGrad over full training, leaning on its average to cut noise without per-layer fiddling.
Alex: Earlier you stressed bias correction—did they test it directly?
Sam: On a variational autoencoder, they varied decay rates. Without correction, high decay for sparsity caused instabilities and high loss. Corrected versions stayed stable, matching or beating RMSProp.
Alex: So for sparse stuff, no correction wrecks it?
Sam: Exactly—new gradients take forever to outweigh zeros otherwise. Correction prevents stalls as gradients sparsify late.
Alex: Any tweaks like Adamax?
Sam: Adamax swaps the wobble measure for the max past gradient size—simpler, stable for sparse high-dim data, no square roots needed.
Alex: So pulling it all together, what makes Adam a game-changer?
Sam: Smart step-sizing via signal-to-noise from gradient moments—bold for clear directions, cautious for noisy—plus bias correction to dodge zero-start stalls. It blends rivals' strengths for sparse glitches and shifting targets, powering fast training from digits to images with minimal babysitting.
Alex: Yeah, and that trust region keeps things safe. But does it always win?
Sam: The evidence points to Adam shining in noisy, high-dim setups, with theory and experiments backing it. For best results, average parameters over time since the last step's noisy. It's robust but watch wild gradients in conv nets; Adamax shows promise.
Alex: I'm struck by how this 2015 optimizer still runs chatbots and image generators today—explains why training feels less like herding cats. Thanks, Sam—this was eye-opening.
Sam: My pleasure, Alex. Thanks for listening to ResearchPod.